A304393 Expansion of Product_{k>0} (1 + Sum_{m>=0} x^(k*2^m)).
1, 1, 2, 2, 5, 5, 8, 10, 17, 19, 27, 33, 48, 56, 76, 92, 126, 146, 192, 228, 298, 352, 444, 528, 667, 783, 969, 1145, 1414, 1658, 2017, 2365, 2878, 3352, 4027, 4703, 5634, 6548, 7773, 9033, 10705, 12381, 14573, 16857, 19790, 22800, 26631, 30655, 35723, 41005
Offset: 0
Keywords
Examples
n | Partitions of n in which each part occurs a power of 2 (cf. A000079) of times --+------------------------------------------------------------------------------ 1 | 1; 2 | 2 = 1+1; 3 | 3 = 2+1; 4 | 4 = 3+1 = 2+2 = 2+1+1 = 1+1+1+1; 5 | 5 = 4+1 = 3+2 = 3+1+1 = 2+2+1; 6 | 6 = 5+1 = 4+2 = 4+1+1 = 3+2+1 = 3+3 = 2+2+1+1 = 2+1+1+1+1; 7 | 7 = 6+1 = 5+2 = 5+1+1 = 4+3 = 4+2+1 = 3+3+1 = 3+2+2 = 3+2+1+1 = 3+1+1+1+1;
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+add(b(n-i*2^j, i-1), j=0..ilog2(n/i)))) end: a:= n-> b(n$2): seq(a(n), n=0..60); # Alois P. Heinz, May 13 2018 -
Mathematica
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i-1] + Sum[b[n-i*2^j, i-1], {j, 0, Floor@Log2[n/i]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Feb 14 2023, after Alois P. Heinz *)
Comments